If \(err_{n+1} \le 1\) , then the computed solution \(u_{n+1}\)  is accepted and advanced. The next time step is computed with \({opt}\)  as step-size. If \(err_{n+1} > 1\) , then step is rejected and the computations are repeated with the new step-size \({opt}\)  . We chose a maximal step-size increase factor of \(facmax = 5\)  to limit large increases in the step-size. In computing the approximate solution immidiately following a step-rejection, we set \(facmax = fac = 0.9\)  to prevent an infinite loop that we observed on rare occasions - similar strategies are employed by [1]}, [2]}, [3]}.
